Finite Group Databasebeta

← #30639644

Subgroups — #30639644

Order 16256.

276 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523238C211yesmaximal, characteristic
2C2×#16861144C211yesmaximal, characteristic
3#22549884C211yesmaximal, characteristic
4C2×#16861128C211yesmaximal, characteristic
5C2×#16861127C211yesmaximal, characteristic
6#22549832C211yesmaximal, characteristic
7#419×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523238—21no—
10C2×#14523238—21no—
11C2×#14523238—22no—
12C2×#14523238C414yescomplemented
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15C22×C127⋊C8C2211yescharacteristic
16#16861144—22no—
17#16861132—22no—
18#16861128—22no—
19#16861127—22no—
20#16861120—21no—
21C4×C127⋊C8—21no—
22C22×Q8×C127C2211yescharacteristic
23C2×M16×C127C2211yescharacteristic
24#117×C127C2211yescharacteristic
25#103×C127—22no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254C2×C412yes—
32C22×Dic127—21no—
33#14523238—24no—
34#14523238—24no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38#14523232D412yes—
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C2×C127⋊C8C2×C412yes—
42C2×C127⋊C8D411yescharacteristic
43C2×C127⋊C8D411yescharacteristic
44C2×C127⋊C8D412yes—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—21no—
50C2×Q8×C127C2×C412yes—
51C22×C4×C127—21no—
52C22×C4×C127C2311yescharacteristic
53M16×C127—21no—
54M16×C127—21no—
55C2×C8×C127—21no—
56C4⋊C4×C127D412yes—
57C22⋊C4×C127—21no—
58C42×C127—21no—
59Dic254—22no—
60Dic254—22no—
61Dic254—41no—
62Dic254—41no—
63C2×Dic127—22no—
64C2×Dic127—41no—
65C2×Dic127—41no—
66C127⋊C8—22no—
67C127⋊C8—41no—
68C127⋊C8C22⋊C412yes—
69C127⋊C8C22⋊C412yes—
70C23×C127#4111yescharacteristic
71Q8×C127—22no—
72Q8×C127—22no—
73Q8×C127—41no—
74Q8×C127—41no—
75C2×C4×C127—22no—
76C2×C4×C127—22no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127C22×C411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C8×C127—41no—
86C22×C127—22no—
87C22×C127C4×D411yescharacteristic
88C22×C127#12111yescharacteristic
89C22×C127#12411yescharacteristic
90C4×C127—42no—
91C4×C127—42no—
92C4×C127C2×C22⋊C411yescharacteristic
93C4×C127C4×D411yescharacteristic
94C4×C127#12111yescharacteristic
95C4×C127#12711yescharacteristic
96Dic127—42no—
97C2×C127—21no—
98C2×C127—21no—
99C2×C127#39011yescharacteristic
100C2×C127#47312yes—
101#1979—1271nomaximal, Sylow
102C127#197911yescharacteristic, complemented, Sylow
103C22×Q16—1271no—
104C2×#108—1271no—
105#419—1271no—
106#419D12711yescharacteristic
107C2×#101—1271no—
108C2×#101—1271no—
109#405—1271no—
110C22×Q8—1271no—
111C22×Q8C2×D12711yescharacteristic
112C2×Q16—1274no—
113C2×Q16—2541no—
114C2×Q16—2541no—
115C2×Q16—2542no—
116C2×M16—1271no—
117C2×M16—1271no—
118C2×M16C2×D12711yescharacteristic
119C22×C8—1271no—
120#117C2×D12711yescharacteristic
121#108—2542no—
122#103—22no—
123#103—2542no—
124#101—2542no—
125#101—2542no—
126#97—2541no—
127C4×C8—2541no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—1272no—
134C2×Q8—2541no—
135C2×Q8—2541no—
136C2×Q8—2541no—
137C2×Q8—2541no—
138C2×Q8—2541no—
139C2×Q8C4×D12712yes—
140C22×C4—21no—
141C22×C4—2541no—
142C22×C4C22×D12711yescharacteristic
143Q16—2544no—
144Q16—2544no—
145M16—21no—
146M16—21no—
147M16—1272no—
148M16—2541no—
149M16—2541no—
150M16—2541no—
151C2×C8—21no—
152C2×C8—1272no—
153C2×C8—1271no—
154C2×C8—1271no—
155C2×C8—1272no—
156C2×C8—2541no—
157C2×C8—2541no—
158C4⋊C4C22⋊D12712yes—
159C22⋊C4—21no—
160C42—21no—
161C23#1452324711yescharacteristic
162Q8—22no—
163Q8—22no—
164Q8—41no—
165Q8—41no—
166Q8—5081no—
167Q8—5081no—
168Q8—2542no—
169Q8—2542no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—5081no—
175C2×C4—5081no—
176C2×C4—2542no—
177C2×C4C2×C4×D12711yescharacteristic
178C2×C4D4×D12711yescharacteristic
179C2×C4D4×D12711yescharacteristic
180C2×C4D4×D12711yescharacteristic
181C2×C4C2×C22⋊D12711yescharacteristic
182C2×C4#1452324811yescharacteristic
183C8—41no—
184C8—1272no—
185C8—1272no—
186C8—5081no—
187C8—2542no—
188C22—22no—
189C22(C22⋊D127)⋊C411yescharacteristic
190C22#1686118111yescharacteristic
191C22#1686119511yescharacteristic
192C4—42no—
193C4—42no—
194C4—5082no—
195C4C22⋊C4×D12711yescharacteristic
196C4C4×C22⋊D12711yescharacteristic
197C4#1686118211yescharacteristic
198C4#1686120511yescharacteristic
199C2—21no—
200C2—21no—
201C2#2254977611yescharacteristic
202C2#2255014912yes—
subgroupG/H
centerC22#16861181
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#419×C127C2
socleC22×C127#121
prime radicalC23×C127#41
cosocleC22×C4×C127C23
hypercenter#419D127
hypocenterC127#1979
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
12. C2×#14523238C4semidirect
102. C127#1979semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#145232382
C2×#168611442
#225498842
C2×#168611282
C2×#168611272
#225498322
#419×C1272
#1979127